Samantha J. Riesenfeld

Ryan A. Robinett, Lorenzo Orecchia, Samantha J. Riesenfeld

We present a framework for efficiently approximating differential-geometric primitives on arbitrary manifolds via construction of an atlas graph representation, which leverages the canonical characterization of a manifold as a finite collection, or atlas, of overlapping coordinate charts. We first show the utility of this framework in a setting where the manifold is expressed in closed form, specifically, a runtime advantage, compared with state-of-the-art approaches, for first-order optimization over the Grassmann manifold. Moreover, using point cloud data for which a complex manifold structure was previously established, i.e., high-contrast image patches, we show that an atlas graph with the correct geometry can be directly learned from the point cloud. Finally, we demonstrate that learning an atlas graph enables downstream key machine learning tasks. In particular, we implement a Riemannian generalization of support vector machines that uses the learned atlas graph to approximate complex differential-geometric primitives, including Riemannian logarithms and vector transports. These settings suggest the potential of this framework for even more complex settings, where ambient dimension and noise levels may be much higher.

Ryan A. Robinett, Sophia A. Madejski, Kyle Ruark et al.

Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach.